Narrow Results

On the basis of the application in the Bose–Einstein condensation, we investigate a (3 + 1)-dimensional Gross–Pitaevskii equation with distributed time-dependent coefficients. With the aid of the Kadomtsev–Petviashvili hierarchy reduction method, we construct the $N$th-order rogue-wave solutions in terms of the Gram determinant by introducing appropriate constraints. Using different coefficients for the diffraction $\beta (t)$ and gain/loss $\gamma (t)$, we demonstrate the behaviors of the first- and second-order rogue waves by analytical and graphical means. We find that only if $\gamma (t)=3\beta (t)$, the rogue waves appear on the constant backgrounds; otherwise, the heights of the backgrounds change as time goes on. With the different choices of $\beta (t)$ and $\gamma (t)$, the long-live, rapid-reducing and periodic rogue waves are discussed. The separated and aggregated second-order rogue waves are also shown on the constant and periodical backgrounds.